# Asymptotic Properties of the Maximum Likelihood Estimator in Regime Switching Econometric Models

## TL;DR

This paper proves the asymptotic normality and consistency of the maximum likelihood estimator in complex Markov regime switching models, including those with zero transition probabilities and regime-dependent densities.

## Contribution

It establishes the asymptotic properties of the MLE for a broad class of Markov regime switching models previously lacking such theoretical validation.

## Key findings

- MLE is asymptotically normal in these models
- Consistency of the covariance matrix estimator is proven
- Applicable to models with zero transition probabilities and regime-dependent densities

## Abstract

Markov regime switching models have been widely used in numerous empirical applications in economics and finance. However, the asymptotic distribution of the maximum likelihood estimator (MLE) has not been proven for some empirically popular Markov regime switching models. In particular, the asymptotic distribution of the MLE has been unknown for models in which some elements of the transition probability matrix have the value of zero, as is commonly assumed in empirical applications with models with more than two regimes. This also includes models in which the regime-specific density depends on both the current and the lagged regimes such as the seminal model of Hamilton (1989) and switching ARCH model of Hamilton and Susmel (1994). This paper shows the asymptotic normality of the MLE and consistency of the asymptotic covariance matrix estimate of these models.